0.S by A1,Th7;
end;
hence thesis by Th7;
end;
definition
let S;
let x be Point of S;
let IT be sequence of S;
attr IT is x-convergent means
:Def4:
IT is convergent & lim IT = x;
end;
theorem Th18:
for X be RealNormSpace for seq be sequence of X holds seq is
constant implies ( seq is convergent & for k be Element of NAT holds lim seq =
seq.k )
proof
let X be RealNormSpace;
let seq be sequence of X;
assume
A1: seq is constant;
then consider r be Point of X such that
A2: for n be Nat holds seq.n=r by VALUED_0:def 18;
thus
A3: seq is convergent by A1,LOPBAN_3:12;
now
let k be Element of NAT;
now
let p be Real such that
A4: 0